Proof sketch ·[Fact灯Q(p)= min{s-rank()):purifies p} -S-rank:Schmidt-rank. Now given Py =(Cx,Dy),let A and B share |〉=∑=1lx)A1vxi)A2☒ly）B1Wy,i)B2 -)ith-row ofCxwyi):ith-column ofDy. Measuring registers (A1,B1): Pr[x,y]=∑，j=1 VxiVxj)(Wy,iwy,）=(Cx,Dy〉 o Similar for the other directionProof sketch • [Fact] 𝑄 𝜌 = min{S−rank(|𝜓〉): |𝜓〉 purifies 𝜌} – S-rank: Schmidt-rank. • Now given 𝑃𝑥𝑦 = 〈𝐶𝑥,𝐷𝑦〉, let A and B share 𝜓 = σ𝑖=1 𝑟 𝑥 𝐴1 𝑣𝑥,𝑖 𝐴2 ⊗ 𝑦 𝐵1 𝑤𝑦,𝑖 𝐵2 – 𝑣𝑥,𝑖 : 𝑖 𝑡ℎ -row of 𝐶𝑥, 𝑤𝑦,𝑖 : 𝑖 𝑡ℎ -column of 𝐷𝑦. • Measuring registers (𝐴1,𝐵1): Pr 𝑥, 𝑦 = σ𝑖,𝑗=1 𝑟 〈𝑣𝑥,𝑖 𝑣𝑥,𝑗 〈𝑤𝑦,𝑖 𝑤𝑦,𝑗 = 𝐶𝑥,𝐷𝑦 • Similar for the other direction